Spectral problems for matrix pencils. Methods and algorithms. II.

This is the second part of the paper by V. B. Khazanov and V. N. Kublanovskaya 'Spectral problems for matrix pencils. Methods and algorithms. Γ. A review of methods and algorithms for solving spectral problems for regular polynomial matrix pencils is presented. Main attention is paid to those methods which are constructed without exploiting the connection between polynomial pencils and linear accompanying pencils or using this connection only implicitly. The paper provides theoretical background for the construction methods to solve problems not only for regular pencils but also for singular pencils although methods and algorithms for the latter are to be considered in the third part of the paper. The present paper is the second part of the paper by V. B. Khazanov and V. N. Kublanovskaya 'Spectral problems for matrix pencils. Methods and algorithms, Γ [54]. The paper treates methods and algorithms for solving spectral problems for regular polynomial matrix pencils. The trivial approach to the solution of problems for polynomial pencils of degree s > 1 consists in solving similar problems for a linear accompanying pencil. In this case, two options arise: either to pass explicitly to a linear problem of a greater size or to exploit implicitly the underlying inter-relations by processing only matrices involved in the original pencil. The first way is reasonable when the pencil degree .s is not too large and the accompanying pencil allows for the size reduction. The thorough description of methods to solve spectral problems for regular pencils by passing explicitly to a linear pencil is given in the review [21]. As to methods from this group designed for the solution of spectral problems for general polynomial pencils of degree s > 1 see [6,18,24,55,66,80,86]. Here, we will consider those methods which either use accompanying pencils only implicitly or do not refer to them at all. The paper consists of two chapters. The first one presents theoretical background underlying the construction of methods. The main attention is paid to different types of accompanying pencils and to the relationship between their spectral characteristics and those of polynomial pencils. In our opinion, this will enable the reader to get oriented in a large number of known linearization algorithms for problems with polynomial dependence on the spectral parameter and also to construct new algorithms of this type. The notions of a block eigenvalue and of a matrix solvent are extended to general pencils, and the relationship between spectral characteristics of block quantities and those of the original pencil is described. The second chapter provides a brief review of known methods and algorithms for 468 V. B. Khazanov and V. N. Kublanovskaya solving spectral problems for regular matrix pencils. Papers considered in [21,77] are only outlined here. Methods for the computation of extreme eigenvalues of the pencil A — λΒ with symmetric matrices, which are reviewed and described in detail in the paper [44] and the monograph [68] are not considered here at all. It was not our intention here to provide a comprehensive review of existing methods for the solution of spectral problems. The paper discusses only the main principles of construction of methods and provides references to publications which describe them. The more complete reference list on this subject can be found in the bibliographical indices [21-23]. Throughout the paper we use the same notation as in [54]. 1. THEORETICAL BACKGROUND 1.1. The Smith canonic form. Spectral characteristics of a polynomial pencil An m χ η matrix pencil 0(λ) = €0 + λ€ι + ··· + λ·€» Cs/0 (1.1) of an integer degree s ̂ 1 is said to be either a Α-matrix or a polynomial matrix, i.e. a matrix whose entries belong to the ring C[A] of scalar polynomials. Let The pencils Ο^λ) and Z)2(A) of the same size are said to be equivalent if the matrices Ρ(λ) and β(Α) are unimodular, that is their determinants are independent of λ and non-zero. If the matrices Ρ and Q are independent of λ and non-singular, then the pencils D^A) and Ο2(λ) are said to be strictly equivalent. One of the important characteristics of a matrix pencil is provided by its rank ρ = rank Ώ(λ) defined as the greatest order of those minors of Z)(A) which are not identically equal to zero. If the two pencil dimensions coincide and are equal to the pencil rank, then the pencil is said to be regular. Otherwise, the pencil is said to be singular. The following theorem is valid [26, 29, 57, 58, 83]. Theorem 1.1. An arbitrary Α-matrix D(A) is strictly equivalent to the Smith canonic form P(A)D(A)Q(A) = Σ(λ) = block diag {^(A), . . . , /cp(A), ®vu}. (1.2) Here, fcf(A) are scalar polynomials (invariant factors) satisfying the following conditions: they are monic, i.e. their leading coefficients are equal to unity, and /c f_ j(A) is a divisor of fc^A) for i = 2, . . . , p. Here, Ovu is a zero υ χ u matrix. The integers Ό = m — p and u = n — p are, respectively, the dimensions of the left (row) and of the right (column) null subspaces Nr(D) and NC(D) of the pencil D(A). Divisors of the form (A — AJ' of the invariant factors are said to be elementary divisors of the pencil D(A) corresponding to its finite eigenvalue A^. To each elementary divisor (A — AJ' there correspond a right and a left Jordan chains of vectors x1? . . . , xt and yi9...9yt satisfying the relations Spectral problems for matrix pencils 469 (sub-scripts of the left Jordan chain are numbered in the reverse order), and the systems are inconsistent. The existence of an elementary divisor of the form μ of the pencil Ζ)(μ) = μΟ(μ~) means that the pencil D(X) has an infinite elementary divisor. Then the corresponding right and left Jordan chains of the pencil Ο(λ) are defined to be the Jordan chains of the pencil D(X) corresponding to its finite elementary divisor μ: Υ C _ χ _ =0 Υ y _ C _ = 0 / = 0 t — 1 (15)